Soft Computing Evaluation Logic

2.7Logic Functions That Use Negation

All GL functions obtained as superposition of idempotent functions are also idempotent. Non‐idempotent functions can be developed using the GCD and negation. In this chapter, we investigate the use of negation as a component in both idempotent and nonidempotent GL functions. In particular, we study the properties and practical use of De Morgan’s duality, complementing and expanding the introduction presented in Sections 2.4.1 and 2.4.4.

2.7.1 Negation and De Morgan’s Duality

Negation is a unary operation, . It determines the truth value of negated proposition and satisfies the boundary conditions and . According to [FOD94], we can differentiate three types of negation:

1. Strict negation: not(x) is continuous and strictly decreasing.
2. Strong negation: not(x) is strict and also involutive, .
3. Standard negation: .

Typical examples of strict negation (but not strong) are and , . An example of strong negation is . For p = 0, this negation becomes standard, ...